Researches on the Distribution of the IVIean Motions of the Asteroids. 3 



where c is a constant depending only on the size of the asteroid, />, 

 tlie density of the particles and F, the resultant relative velocity of 

 the asteroid. The constants c and p are essentially positive. 



2. Let /> be a continuous ftmction of r, so that it may be devel- 

 oped in the convergent series, 



Put 



7 _ a- dp 7, _ 1 a' d'-p 



^ Pq da' '^2 po dar' 



then 



,=4n.,,i^+^r=<i)+ } 



Or developing —— in powers of e, 



(3) />=(O0(1 — ÄTi^ cos ÎU— Ä;ie^sin^'iü + A-2e' cos- ÎU+ ) 



3. The equations for the variations of the elements are^^ 



^= ^==- [Se sin w + T{l + ecos ?o)1 



de Vi — r Va ■ , rrf , cosîo + e \~1 



L \ 1 + e cos ?o/J 



— >Scosîu + T( l+-i |sin?t' 



L \ l+eco&iv/ J 



dt na 



dw^ VT^ 

 dt na 



ds __ 2(1—6^) çy e" dv3 



dt 7ia(l + e cos iv) 1 + VI — e"^ dt 



where -iss and e are the mean longitude of perihelion and the mean 

 longitude at the epoch. Since V contains the 1st power of e as 

 factor, S and T will contain e'' as factor. Hence, if we confine 

 ourselves to the order of e''^^ in the development of the differential 

 coefficients, we may neglect e^ terms in the coefficients of S and 

 T. Thus, 



1) Tisserand, Mécanique Céleste, I p. 433 and IV p. 218. 



